About the robustness of adaptive feedback linearization controller for input perturbed uncertain fully-Actuated systems - Nguyen Van Chi

Journal of Science and Technology 54 (2) (2016) 276-285 DOI: 10.15625/0866-708X/54/2/6233 276 ABOUT THE ROBUSTNESS OF ADAPTIVE FEEDBACK LINEARIZATION CONTROLLER FOR INPUT PERTURBED UNCERTAIN FULLY-ACTUATED SYSTEMS Nguyen Van Chi1, *, Nguyen Hien Trung1, Nguyen Doan Phuoc2, 1Thai Nguyen University of Technology, 3/2 Street Tich Luong, Thai Nguyen 2Hanoi University of Science and Technology, 1 Dai Co Viet Street, Hanoi *Email: ngchi@tnut.edu.vn Received: 21 May 2015; Accepted for publication: 25 November 2015 ABSTRACT The paper proposes a theorem to assert the arbitrarily good robustness of the fully actuated mechanical system controlled by the adaptive feedback linearization controller. The fully actuated system to be controlled is considerately perturbed by input disturbances and contains constant uncertain parameters in its Euler-Lagrange forced model. It is shown in this paper that independent of input disturbances of the adaptive feedback linearization controller with appropriately chosen parameters will drive the output of controlled systems to the desired trajectory for any arbitrary precision. The adaptive controller is applied to the two-link planar elbow arm robot with unknown mass of the end-effector of second link and input torque noises caused by the viscous friction forces and Coulomb friction terms. Simulation results show that the arbitrary precision of the tracking errors always is guaranteed. Keywords: feedback linearization, robust adaptive feedback control, uncertain systems, Euler- Lagrange forced model. 1. INTRODUCTION The uncertainness of fully actuated mechanical systems, which is commonly described by an Euler-Lagrange forced model as follows [1]: ( , ) ( , , ) ( , )M q q C q q q g q uθ θ θ+ + =ɺɺ ɺ ɺ (1) is understood that the q - dimensional vector and θ of model parameters are constant but unknown, which is however linear dependent on the system in the sense of: 0( , ) ( , , ) ( , ) ( , , ) ( , , )M q q C q q q g q F q q q F q q qθ θ θ θ+ + = +ɺɺ ɺ ɺ ɺ ɺɺ ɺ ɺɺ (2) In the Euler-Lagrange model given above the n dimensional vector q is called the vector of configuration variables, u is the n dimensional vector of n control inputs, ( , ) n nM q θ ×∈R About the robustness of adaptive feedback linearization controller... 277 is the inertia matrix, which is symmetric and positive definite, and ( , , ) n nC q q θ ×∈ɺ R is the centripetal and coriolis forces corresponding matrix. To tracking control for this uncertain system in the sense, that the tracking error r e q q= − has to be bounded for all 0t ≥ and asymptotically convergence to the origin, where ( ) r q t is any desired trajectory, the adaptive controller presented in [1, 2]: ( ) 1 2 with ( , ) ( , ) ( , , ) ( , ) T r BF Px x col e e u M q q K e K e C q q q g q θ θ θ θ  = =   = + + + +   ⌢ɺ ɺ ⌢ ⌢ ⌢ ɺɺ ɺ ɺ ɺ (3) is widely admitted to be an effective solution [1 - 3], where the 2n n× matrix B is defined by: 1( , )B M q θ− Θ  =       ⌢ in which Θ is the n n× zeros matrix, 1 2, K K are any two selected n n× matrices such that the 2 2n n× matrix: 1 2 I A K K Θ  =   − −  with the n n× identity matrix I , will be Hurwitz, and the symmetric positive definite 2 2n n× matrix P is the solution of the Lyapunov equation: ( )12 TA P PA Q+ = − where Q is also an arbitrarily chosen symmetric positive definite 2 2n n× matrix. In many references the adaptive controller (3) is referred to as the adaptive feedback linearization controller. Furthermore, as it is shown in [2 - 4], for the control problem of input perturbed uncertain systems: ( , ) ( , , ) ( , )M q q C q q q g q u nθ θ θ+ + = +ɺɺ ɺ ɺ (4) where ( )n t is the vector of input noises, which is assumed to be bounded: sup ( ) t n tδ = the q -feedback adaptive feedback linearization controller (3) given above always drives the tracking error ( , )x col e e= ɺ of the closed loop system depicted in Fig. 1 asymptotically to the neighborhood O of the origin defined by: 2 min ( ) n PBx x Q δ λ    = ∈ ≤     O R (5) where min ( )λ ⋅ denotes the minimal eigenvalue and ⋅ the norm of a matrix. The neighborhood O is also referred as the attractor of closed loop systems. The smaller this attractor is, the better tracking performance of the system is. Nguyen Van Chi, Nguyen Hien Trung, Nguyen Doan Phuoc 278 Figurre 1. Structure of the closed loop system obtained by using the adaptive feedback linearization controller (3). Since the feedback linearization controller (3) contains in it some freely selected parameters such as two matrices 1 2, K K and the symmetric positive definite matrix P , the robust tracking performance defined in the equation (5) above of the closed loop system depicted in 0 could be evidently improved further, if these parameters have been suitably chosen. And this paper will present a methodology to determine matrices 1 2, , K K P for adaptive feedback linearization controller (3) so that the tracking behaviour of the obtained closed loop system satisfies any desired arbitrarily small attractor O . 2. MAIN RESULT Also according to the suggestion of [1], both matrices 1 2, K K of the feedback linearization controller (3) could be chosen diagonally: 1 1 2 2( ), ( ), 1,2, ,i iK diag k K diag k i n= = = and appropriately the matrix Q of the form: 2 2 1 1 2 2 2 1 2 1 ( ) ( ) i i i K diag k Q K K diag k k    Θ Θ    = =    Θ − Θ −    (6) In this circumstance the matrix A is Hurwitz if and only if: 2 1 2 10, i i ik k k> > for all 1,2, ,i n= and the Lyapunov equation has the following unique solution: 1 2 1 1 2 2K K K P K K   =     (7) which is obviously symmetric and positive definite. Moreover, it is easily to recognize from the equation (5), that the measure of O defined as follows: ( ) , max x y m x y= −O for all ,x y ∈O is an intuitive value to appreciate the robustness of the closed loop system. The smaller ( )m O is, the better robustness of the system is. q , , r r r q q qɺ ɺɺ θ n Controlled system (1) Linearization Controller (3) About the robustness of adaptive feedback linearization controller... 279 Theorem: For any given 0ε > always exits two matrices 1 2, K K such that the proposed q - feedback dynamic controller (3) satisfies the desired robustness: ( )m ε≤O (8) Proof: Chosen 1 2, K K diagonally with: 1 ( ), 1K diag k k= > and 2 ( ), 2K diag ak a= > (9) as well as Q from the structure (6), then there are obtained: ( ) 1 1 2 1 1 1 21 11 2 2 2 max , i i i K K K K M PB k k K K M K M γ − − −  Θ    = = ≤          ⌢ ⌢ ⌢ and 2 2 1 1 2 2 2 1 2 1 ( ) ( ) i i i K diag k Q K K diag k k    Θ Θ    = =    Θ − Θ −    ⇒ ( )2 2min 1 2 1( ) min , i i i i Q k k kλ = − where M ⌢ is the short expression of the matrix ( )( , ) ( , )ijM q m qθ θ=⌢ ⌢ and: 1 1 1 max ( , ) m ij i m j M m qγ θ− ≤ ≤ = = = ∑ ⌢ ⌢ Hence, it deduces: ( ) ( ) ( ) ( ) ( ) 2 2 2 2 2 2 min 2 2 2 2 2 2 2 max , ( ) min , min , min , min , ( 1) i i i i i k akPB ak Q k a k k k a k k ak ak a kk a k k k a k γδδ γδ λ γδ γδ γδ = = − − ≤ = = − − and from which to find out: lim 0 k a k γδ →∞ = Therefore, by any given 0ε > always exists a sufficiently large number 0k > such that: ( ) am k γδ ε≤ <O which affirms the rightness of Theorem. ♦ 3. NUMERICAL EXAMPLE To illustrate the proposed theorem it is considered hereafter a two-link planar elbow arm robot (Fig. 2), which is now additionally perturbed by input noises 1 2( , )Tn n n= and described by the uncertain Euler-Lagrange forced model (4) with the following parameters [1]: Nguyen Van Chi, Nguyen Hien Trung, Nguyen Doan Phuoc 280 1 1 1 2 2 2 2 2 2 1 2 1 2 2 2 1 2 2 2 2 2 1 2 2 2 1 2 2 2 1 2 2 2 1 2 1 2 1 1 , , ( ) 2 cos ( cos )( , ) ( cos ) 2 sin sin( , , ) sin 0 ( ) cos( , ) n q u n n m l l l l l l l M q l l l l l l l l C q q l l m gl g g q ϕ τ ϕ τ θ θ θ ϕ θ ϕ θ θ ϕ θ θ ϕ ϕ θ ϕ ϕ θ θ ϕ ϕ θ ϕ θ θ       = = =             + + + +  =  +  − −  =     + + = ɺ ɺ ɺ ɺ ɺ ɺ 2 1 2 2 1 2 cos( ) cos( ) l gl ϕ ϕ θ ϕ ϕ +    +  (10) where 1 2( , )Tτ τ τ= is the input vector, in which the torque 1τ produces the angular motion 1ϕ and the torque 2τ produces the angular motion 1ϕ of robot arms. Figure 2. The controlled system is a two-link planar elbow arm robot. Now, the adaptive controller (3) is applied to the arm robot in Fig. 2 for tracking problem of the angles and the velovities of two links, by using two diagonal matrices 1 2, K K suggested in (9) with 2a = : 1 2 0 2 0 , 0 0 2 k k K K k k     = =        ⇒ 2 1 2 2 2 0 0 2 k K K k    =     and 2 21 2 1 1 2 4 0 0 2 0 4 0 0 2 0 0 0 2 k k K K K k kP K K k k k k         = =            the feedback linearization controller (3) for the controlled system (10) with parameters: 2 1 29.8 / , 1 , 2.5 , 0.5 , 0.5g m s m kg kg l m l mθ= = = = = becomes ( )( )22 1 2 1 22 2 1 ˆ0 , 0 , - , - / ˆ( ) e l f af af c b f P ec b l θ θ θ   = + +   +   ⌢ɺ ɺ (11) where 1 2 1 2, ( , ) , ( , )T r r Tr re q q q qϕ ϕ ϕ ϕ= − = = denotes the tracking deviation and − 1u is the torque which produces the angle 1ϕ − 2u is the torque which produces the angle 2ϕ − θ is the mass which is not exactly measurable. − 9.81 2g m s= is the acceleration of gravity 2ϕ 2l g θ m 1l 1ϕ About the robustness of adaptive feedback linearization controller... 281 1 2 2 12 ˆ ˆ 2 ( )2 ˆ ˆ r c b a d d m h f u q ke ke e fa l θ θ ϕ ϕ θ θ ϕ θθ θ  +   − − + + +  = + + +      +   ⌢ ⌢ ɺ ɺ ɺɺ ɺ ⌢ ɺ (12) with 2 2 1 2 2 2 1 2 1 cosa l l l b l a c ml ϕ= + = + = , 1 2 2 2 1 2 1 2 sin sin d l l e l l ϕ ϕ ϕ ϕ = = ɺ ɺ ɺ , 2 1 2 1 1 cos( ) cos f gl h gl ϕ ϕ ϕ = + = and 1 1 2 1 2 2 2 1 2 2 1 2f b a d d h f f a l e ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ = + − − + + = + + + ɺɺ ɺɺ ɺ ɺ ɺɺ ɺɺ ɺ Figure 3 and Fig. 4 depict angle and velocity simulation results obtained with 3, 10k k= = and 30k = respectively. In this simulation, the input noises applying in two links are considered to be depend on velocities of the links as below: 1 1 1 1 2 2 2 2 2 2 1 1 ( ) 3 ( ) 0.5 ( ) 5 (1,1) ( ) 5 ( ) 0.32 ( ) 5 (1,1) n t sign sign rand n t sign sign rand ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ = + + = + + ɺ ɺ ɺ ɺ ɺ ɺ ɺ ɺ ɺ ɺ (13) The Fig. 4 shows that the response angles of the robot arm track to the set points after the transient period in 7.5 seconds. There is 0.113 rad of maximum angle errors which reduce to 2.55×10-3 rad by using 10k = and 5.3×10-4 rad by 30k = as showing in the Fig. 5. The more k increases, the more angle errors and velocity errors reduce. Figure 3. Desired angles, simulated angles (a) and desired velocities, simulated velocities (b) of first link and second link with 3, 10k k= = and 30k = . 0 5 10 15 20 25 30 0 0.5 1 1.5 time(s) An gl e of fir st lin k ( ra d) 0 5 10 15 20 25 30 -1 0 1 time (s) An gl e of se co n d lin k (ra d) k=3 setpoint k=10 k=30 k=3 k=10 k=30 setpoint b) a) 0 5 10 15 20 25 30 -0.5 0 0.5 1 1.5 2 time (s) Ve lo ci ty of fir st lin k (ra d/ s) 0 5 10 15 20 25 30 -2 -1 0 1 2 time(s) Ve lo ct y of se co nd lin k (ra d/ s) k=3 setpoint k=10 k=30 k=3 setpoint k=10 k=30 Nguyen Van Chi, Nguyen Hien Trung, Nguyen Doan Phuoc 282 Figure 4. Angle errors(a) and velocities errors (b) of first link and second link with 3, 10k k= = and 30k = . Figure 5. Input noise of the first link (a) and Input noise of the second link (b) with 3, 10k k= = and 30k = . 0 5 10 15 20 25 30 -20 0 20 40 60 0 5 10 15 20 25 30 -50 0 50 0 5 10 15 20 25 30 -20 0 20 40 60 To rq ue of 1s t l in k(N . m ) 0 5 10 15 20 25 30 -20 0 20 40 To rq u e of 2n d lin k(N . m ) 0 5 10 15 20 25 30 0 100 200 time(s) 0 5 10 15 20 25 30 0 50 100 time(s) k=3 k=3 k=10 k=10 k=30 k=30 Figure 6. The torques apply to the first link and second link with 3, 10k k= = and 30k = . In the Fig. 6 there are input torques computed by the adaptive controller to get the tracking of the links, the maximum amplitudes of input torques is 60 N.m with 3k = and 200 N.m with 30k = . b) a) 0 5 10 15 20 25 30 0 5 10 15 time(s) In pu t n oi se of fir st lin k (N . m ) k=3 k=10 k=30 0 5 10 15 20 25 30 -15 -10 -5 0 5 10 time(s) In pu t n oi se of se co n d lin k (N . m ) k=3 k=10 k=30 b) a) 0 5 10 15 20 25 30 0 0.5 1 1.5 2 An gl e er ro r of 1s t l in k(r ad ) k=3 k=10 k=30 0 5 10 15 20 25 30 -0.5 0 0.5 1 1.5 2 time(s) An gle er ro r of 2n d lin k(r ad ) k=3 k=10 k=30 0 5 10 15 20 25 30 -2 -1.5 -1 -0.5 0 0.5 Ve lo ci ty er ro r of 1s t l in k(r ad /s ) k=3 k=10 k=30 0 5 10 15 20 25 30 -1 0 1 2 time(s) Ve lo ci ty er ro r of 2n d lin k(r ad /s ) k=3 k=10 k=30 About the robustness of adaptive feedback linearization controller... 283 0 5 10 15 20 25 30 0 2 4 6 8 time(s) Th et a ha t k=3 k=10 k=30 Figure 7. The adaptive parameter ˆθ with 3, 10k k= = and 30k = . 0 5 10 15 20 25 30 -1 0 1 2 An gl e of 1s t l in k(r ad ) 0 5 10 15 20 25 30 -1 0 1 2 An gl e of 2n d lin k(r ad ) 0 5 10 15 20 25 30 -1 0 1 2 time(s)V el oc ity of 1s t l in k(r ad /s ) 0 5 10 15 20 25 30 -1 0 1 2 time(s)V el oc ity of 2n d lin k(r ad /s ) m2=1kg m2=2.5kg m2=5kg setpoint m2=2.5kg setpoint m2=5kg m2=1kg m2=1kg setpoint m2=2.5kg m2=5kg m2=1kg setpoint m2=2.5kg m2=5kg Figure 8. The responses of angle and velocity with changing of the mass of the end-effector. 0 5 10 15 20 25 30 0 2 4 6 8 time(s) Th et a ha t m2=1kg m2=2.5kg m2=5kg Figure 9. The adaptive parameter ˆθ with changing of the mass of the end-effector. The adaption of the parameter ˆθ with 3, 10k k= = and 30k = are depicted in the Fig. 7. It changes strongly when the arm robot is effected by input noises and it reached to the real value of the mass of the end-effector when the input noises are zero. The Fig. 8 shows that the angle and velocity responses by changing of the mass at the end-effector with 2 1m kg= , 2 2.5m kg= and 2 5m kg= are not quite different. It means that the influence of 2m to the angles and velocities has been attenuated by the adaptation of ˆθ as showing in Fig. 9. Finally, all obtained simulation results above have concluded that any desired robustness for the control of systems with unknown parameters and input noises (4), will be always satisfied with the feedback linearization controller (3). Nguyen Van Chi, Nguyen Hien Trung, Nguyen Doan Phuoc 284 3. CONCLUSIONS This paper refers to robustness of the fully actuated mechanical system which is considered by Euler – Lagrange forced model with input disturbances and contains constant uncertain parameters. By giving and proofing a theorem we conclude that the outputs of the system controlled by the adaptive feedback linearization controller will track to the desired trajectories for any arbitrary precision with appropriately chosen controller parameters. The adaptive controller is proposed in this paper not only keeps the tracking of the outputs in the presence of the uncertain parameters but also attenuates the influence of the input noises to the system. For more details, the adaptive controller is applied to the tracking problem of the two-link planar elbow arm robot with unknown mass of the end-effector and the influences of the noises to the input torques, the simulation results show that we can get the arbitrary precision of the angles and velocities of the links. The proof of the convergence of adaptive parameters to real values of unknown parameters and applying this control method to the practice are our further researches. REFERENCES 1. Levis F.L; Dawson D.M. and Abdalla C.T. - Robot manipulator control. Theory and practice, Marcel Decker Inc, (2006). 2. Chi N.V and Phuoc N.D. - Adaptive tracking control based on disturbance attenuation and ISS stabilization of Euler-Lagrange nonlinear in the presence of uncertainty and input noise, Proceedings of IEEE 2th Int. Conference on Electrical Engineering and Automation Control (EEAC2011), Aug. 2011 China, pp. 3698-3071, (2011) 3. Quan N.T. and Phuoc N.D. - Robust and adaptive control of Euler-Lagrange systems with an attractor independent of uncertainties, Proceedings of IEEE 12th Int, Conference on Control, Automation and Systems ICCAS 2012, IEEE Catalog Number CFP1210D-CDR, (2012) pp. 1309-1312. 4. Chi Nguyen Van, Phuoc Nguyen Doan -Adaptive ISS Stabilization Tracking Control Of Nonlinear Systems In The Presence Of Uncertainty And Input Noise With Guaranteed Tracking Errors, Journal of Computer Science and Cybernetics 29 (2) (2013) 132-141. TÓM TẮT NGHIÊN CỨU TÍNH BỀN VỮNG CỦA BỘ ĐIỀU KHIỂN TUYẾN TÍNH HÓA PHẢN HỒI THÍCH NGHI CHO CÁC HỆ CƠ KHÍ ĐẦY ĐỦ CƠ CẤU CHẤP HÀNH CÓ NHIỄU BẤT ĐỊNH ẢNH HƯỞNG ĐẾN ĐẦU VÀO Nguyễn Văn Chí1, Nguyễn Hiền Trung1, Nguyễn Doãn Phước2 1Đại học Kỹ thuật công nghiệp Thái Nguyên, Đường 3/2 – Tích Lương, TP Thái Nguyên 2Đại học Bách khoa Hà Nội, Số 1 Đại Cồ Việt, Hà Nội *Email: ngchi@tnut.edu.vn Bài báo này đưa ra một định lí và khẳng định tính bền vững tùy ý cho hệ cơ khí đầy đủ cơ cấu chấp hành có các tham số bất định và nhiễu đầu vào mô tả dưới dạng mô hình Euler- Lagrange được điều khiển bằng bộ điều khiển tuyến tính hóa phản hồi thích nghi. Bộ điều khiển About the robustness of adaptive feedback linearization controller... 285 tuyến tính hóa phản hồi thích nghi với các tham số được chọn một cách phù hợp sẽ điều khiển đầu ra của hệ bám theo quỹ đạo mong muốn với độ chính xác yêu cầu mà không phụ thuộc vào nhiễu đầu vào. Bộ điều khiển được áp dụng cho hệ robot khuỷu tay hai thanh nối với khối lượng điểm cuối không biết trước và có mô men đầu vào chịu ảnh hưởng của các lực ma sát nhớt và các thành phần ma sát Coulomb. Kết quả mô phỏng cho thấy rằng độ chính xác tùy ý của sai lệch bám quỹ đạo luôn luôn được đảm bảo. Từ khóa: tuyến tính hóa phản hồi, điều khiển phản hồi thích nghi bề vững, các hệ bất định, hệ phi tuyến Euler-Lagrange.